Ordinary least squares: Meaning, Criticisms & Real-World Uses

What Is Ordinary Least Squares?

Ordinary least squares (OLS) is a foundational statistical method used within statistical modeling, primarily for linear regression analysis. It is a technique for estimating the unknown coefficients in a linear regression model by minimizing the sum of the squared differences between the observed values and the values predicted by the model⁹⁹, ¹⁰⁰. These differences are known as residuals or errors⁹⁸. The core idea of OLS is to find the "line of best fit" through a set of data points, such that the vertical distances from each point to the line, when squared and summed, are as small as possible⁹⁷. This makes Ordinary Least Squares a popular choice for understanding the relationship between a dependent variable and one or more independent variables ⁹⁶.

History and Origin

The method of least squares, which forms the basis of Ordinary Least Squares, was developed independently by two prominent mathematicians in the early 19th century. The French mathematician Adrien-Marie Legendre first published the method in 1805 in his work on celestial mechanics, "Nouvelles méthodes pour la détermination des orbites des comètes.". A⁹⁴, ⁹⁵round the same time, the German mathematician Carl Friedrich Gauss claimed to have been using the method since 1795, although he did not publish his findings until 1809 in "Theoria Motus Corporum Coelestium". T⁹², ⁹³here has been historical debate regarding who discovered it first, but Legendre is recognized for the initial publication. Gauss, however, is often credited with a more sophisticated development, linking the method to probability and providing algorithms for computation. T⁹¹he principle was initially applied to problems in astronomy and geodesy, such as predicting comet orbits based on imprecise measurements. M⁹⁰ore details on this historical context can be found in discussions of the method's origins Least squares: Who came up with this idea?.

Key Takeaways

Ordinary Least Squares (OLS) is a statistical method for estimating parameters in linear regression models.
It minimizes the sum of squared errors between observed data points and the regression line.
OLS is widely used for predictive modeling, relationship analysis, and hypothesis testing in various fields, including finance.
*⁸⁹ The method relies on several assumptions about the data to produce reliable and efficient estimates.
Interpreting OLS results involves analyzing coefficients, R-squared, p-values, and diagnostic statistics.

Formula and Calculation

The goal of Ordinary Least Squares is to find the values for the regression coefficients (the slope and intercept for simple linear regression, or multiple slopes for multiple linear regression) that minimize the sum of the squared residuals.

For a simple linear regression model with one independent variable, the equation is typically expressed as:
[
Y = \beta_0 + \beta_1 X + \epsilon
]
Where:

(Y) is the dependent variable
(X) is the independent variable
(\beta_0) is the intercept (the value of Y when X is 0)
(\beta_1) is the slope coefficient (the change in Y for a one-unit change in X)
(\epsilon) is the error term (the residual)

The OLS method seeks to minimize the function:
[
\text{Minimize } \sum_{i=1}^{n} (y_i - \hat{y}i)^2 = \sum{i=1}^{{n} (y_i - (\hat{\beta}_0 + \hat{\beta}_1 x_i))}2
]
Where (y_i) are the observed values, (\hat{y}_i) are the predicted values, and (\hat{\beta}_0) and (\hat{\beta}_1) are the estimated coefficients.

The estimated slope ((\hat{\beta}_1)) and intercept ((\hat{\beta}_0)) can be calculated using the following formulas for simple linear regression:
[⁸⁷](https://library.fiveable.me/linear-modeling-theory-and-applications/unit-2/ordinary-squares-ols-method/study-guide/EblpWZA4oQC3uKFw), ⁸⁸[
\hat{\beta}_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
]
[
\hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x}
]
Here, (\bar{x}) and (\bar{y}) represent the means of the independent and dependent variables, respectively. For multiple linear regression, matrix algebra is used to solve for the coefficients.

⁸⁶## Interpreting the Ordinary Least Squares Results

Interpreting the results of an Ordinary Least Squares regression analysis involves examining several key statistics to understand the model's performance and the relationships between variables.

⁸⁵First, the estimated coefficients (or beta values) indicate the strength and direction of the relationship between each independent variable and the dependent variable. F⁸³, ⁸⁴or example, a positive coefficient for an independent variable suggests that as that variable increases, the dependent variable is expected to increase, assuming all other variables remain constant.

⁸²Second, the R-squared ((R^2)) value is a crucial metric that represents the proportion of the variance in the dependent variable that is explained by the independent variables in the model. R⁸¹-squared values range from 0 to 1, with higher values indicating a better fit of the model to the data. H⁸⁰owever, a high R-squared alone does not guarantee a good model, as it can increase simply by adding more independent variables. T⁷⁹he adjusted R-squared is often preferred as it accounts for the number of predictors in the model.

⁷⁸Third, the p-value for each coefficient assesses its statistical significance. A⁷⁶, ⁷⁷ p-value typically less than 0.05 suggests that the corresponding independent variable has a statistically significant effect on the dependent variable, meaning the observed relationship is unlikely to be due to random chance. S⁷⁴, ⁷⁵imilarly, the F-statistic and its associated p-value evaluate the overall statistical significance of the entire Ordinary Least Squares model, testing whether at least one independent variable has a significant effect on the dependent variable.

⁷², ⁷³Diagnostic plots and tests are also used to check the underlying assumptions of OLS, such as the linearity of the relationship, the independence of errors (no autocorrelation), constant variance of residuals (homoscedasticity), and normality of errors. V⁷¹iolations of these assumptions can affect the reliability of the OLS estimates and their interpretation.

⁷⁰## Hypothetical Example

Imagine an investor wants to understand how a company's advertising expenditure impacts its quarterly sales. They collect data for the past eight quarters, with advertising expenditure in thousands of dollars and sales in millions of dollars.

Quarter	Advertising Expenditure (X)	Sales (Y)
1	2	5
2	3	7
3	4	8
4	5	10
5	6	11
6	7	13
7	8	14
8	9	16

To apply Ordinary Least Squares, the investor would calculate the mean of X ((\bar{x})) and Y ((\bar{y})).
(\bar{x} = (2+3+4+5+6+7+8+9) / 8 = 44 / 8 = 5.5)
(\bar{y} = (5+7+8+10+11+13+14+16) / 8 = 84 / 8 = 10.5)

Next, the sums needed for the slope and intercept formulas are computed:
(\sum (x_i - \bar{x})(y_i - \bar{y}))
(= (2-5.5)(5-10.5) + (3-5.5)(7-10.5) + ... + (9-5.5)(16-10.5))
(= (-3.5)(-5.5) + (-2.5)(-3.5) + (-1.5)(-2.5) + (-0.5)(-0.5) + (0.5)(0.5) + (1.5)(2.5) + (2.5)(3.5) + (3.5)(5.5))
(= 19.25 + 8.75 + 3.75 + 0.25 + 0.25 + 3.75 + 8.75 + 19.25 = 64)

(\sum (x_i - \bar{x})^2)
(= (-3.5)^{2 + (-2.5)}2 + (-1.5)^{2 + (-0.5)}2 + (0.5)^{2 + (1.5)}2 + (2.5)^{2 + (3.5)}2)
(= 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 = 42)

Now, calculate the estimated slope ((\hat{\beta}_1)):
(\hat{\beta}_1 = 64 / 42 \approx 1.52)

And the estimated intercept ((\hat{\beta}_0)):
(\hat{\beta}_0 = 10.5 - 1.52 * 5.5 \approx 10.5 - 8.36 = 2.14)

So, the estimated OLS linear regression equation is:
Sales = 2.14 + 1.52 * Advertising Expenditure

This suggests that for every additional thousand dollars spent on advertising, quarterly sales are expected to increase by approximately $1.52 million. This hypothetical example illustrates how OLS estimates the relationship between two variables. The differences between the actual sales and the sales predicted by this equation are the residuals, which Ordinary Least Squares aims to minimize in their squared sum.

Practical Applications

Ordinary Least Squares is a versatile technique with extensive practical applications in finance and economics. I⁶⁹n quantitative finance, OLS is a fundamental tool for various analyses:

Asset Pricing: OLS is used to model the relationship between an asset's returns and various market factors or economic indicators. For example, the Capital Asset Pricing Model (CAPM) is a well-known application where OLS regression is used to determine the expected return of an asset based on its systematic risk (beta) and the market's expected return.
*⁶⁸ Risk Management: Financial institutions employ OLS to assess and forecast different types of risk, such as credit risk or market risk. By regressing default rates against borrower characteristics, banks can estimate the probability of default for loan applicants. I⁶⁶, ⁶⁷t can also be used to analyze the impact of economic indicators on investment portfolios.
Forecasting Market Trends: Analysts use OLS to build predictive models for various financial metrics, including stock prices, interest rates, and commodity prices. B⁶⁴, ⁶⁵y identifying significant drivers from historical data, OLS helps to inform trading strategies and investment decisions.
*⁶³ Econometrics: In broader economics, OLS is applied to study relationships between macroeconomic variables, such as the impact of GDP growth on unemployment rates (Okun's law) or the effect of monetary policy on inflation.
*⁶² Portfolio Management: OLS can assist in optimizing portfolio allocation by understanding the correlation and covariance between different assets, helping to achieve optimal risk-return profiles.

The process typically involves collecting historical data, specifying the regression model, and then applying OLS to estimate the coefficients that explain the relationship. T⁶¹hese applications highlight the utility of OLS in converting observed values into predicted insights for informed decision-making. F⁶⁰urther insights into its use in financial analysis can be found in discussions of linear regression in quantitative finance Harnessing the Power of Linear Regression in Quantitative Finance.

Limitations and Criticisms

Despite its widespread use and relative simplicity, Ordinary Least Squares (OLS) has several limitations and underlying assumptions that, if violated, can lead to unreliable or inefficient estimates.

One significant limitation is its sensitivity to outliers. B⁵⁸, ⁵⁹ecause OLS minimizes the sum of squared differences, extreme values can disproportionately influence the estimated regression line, pulling it away from the majority of the data points and leading to biased predictions.

⁵⁶, ⁵⁷Another critical aspect relates to its classical assumptions, often referred to as the Gauss-Markov assumptions. These include:

Linearity: The relationship between the independent and dependent variable must be linear in the parameters.
*⁵³, ⁵⁴, ⁵⁵ No Endogeneity: The independent variables must not be correlated with the error term.
*⁵¹, ⁵² Homoscedasticity: The variance of the residuals (errors) must be constant across all levels of the independent variables. I⁴⁷, ⁴⁸, ⁴⁹, ⁵⁰f this assumption is violated (known as heteroscedasticity), OLS estimates remain unbiased but become inefficient, meaning their standard errors are unreliable, which can affect hypothesis testing.
*⁴⁵, ⁴⁶ No Autocorrelation: The errors for different observations must be uncorrelated with each other. T⁴³, ⁴⁴his is particularly relevant in time series data, where errors in one period might be correlated with errors in a subsequent period.
No Multicollinearity: Independent variables should not be highly correlated with each other. H⁴¹, ⁴²igh multicollinearity can make it difficult to determine the individual impact of each independent variable and can lead to unstable coefficient estimates.
*⁴⁰ Normality of Errors (Optional for efficiency, crucial for inference): While OLS does not strictly require normally distributed errors to produce unbiased estimates, satisfying this assumption allows for valid statistical hypothesis testing and the construction of reliable confidence intervals.

³⁷, ³⁸, ³⁹If these assumptions are not met, the Ordinary Least Squares model's results may be misleading or inefficient. F³⁶or instance, if the true relationship between variables is non-linear, applying OLS may lead to inaccurate model results. F³⁴, ³⁵urthermore, OLS can become less effective when dealing with a very large number of features relative to the data points, potentially leading to non-unique solutions. U³³nderstanding these limitations is crucial for correctly applying and interpreting OLS Understanding The Limitations Of Ordinary Least Squares (ols). A comprehensive overview of these assumptions can be found in statistical guides 7 Classical Assumptions of Ordinary Least Squares (OLS) Linear Regression.

Ordinary Least Squares vs. Maximum Likelihood Estimation

Ordinary Least Squares (OLS) and Maximum Likelihood Estimation (MLE) are both methods for estimating parameters in statistical models, but they operate on different principles and have distinct applications.

OLS is a distance-minimizing approach that focuses on finding the regression line that minimizes the sum of the squared differences (residuals) between the observed data and the values predicted by the model. I³¹, ³²t is primarily used for regression analysis and does not inherently require specific assumptions about the probability distribution of the errors to find the estimates.

²⁹, ³⁰In contrast, Maximum Likelihood Estimation (MLE) is a probabilistic approach that seeks to find the parameter values that maximize the likelihood of observing the given dataset. T²⁷, ²⁸his means MLE assumes a specific probability distribution for the error term, most commonly a normal (Gaussian) distribution. M²⁵, ²⁶LE then identifies the parameters that make the observed data most probable under that assumed distribution.

²³, ²⁴Despite their different foundational approaches, OLS can be considered a special case of MLE. If the errors in a linear regression model are assumed to be independently and identically distributed with a normal distribution and a mean of zero, then the OLS estimates for the coefficients will be identical to those obtained through MLE.

¹⁹, ²⁰, ²¹, ²²The choice between OLS and MLE often depends on the specific characteristics of the data and the modeling goals. For large and complete datasets, both methods tend to yield consistent results. However, MLE is generally considered more versatile and applicable to a wider range of models and data types, especially when dealing with smaller samples or censored data.

¹⁸## FAQs

What does "least squares" mean in Ordinary Least Squares?

"Least squares" refers to the core principle of the Ordinary Least Squares method, which is to minimize the sum of the squared differences between the actual observed values and the values predicted by the model. B¹⁶, ¹⁷y squaring the differences (residuals), positive and negative errors do not cancel each other out, ensuring that larger deviations are penalized more heavily.

¹⁵### Why is Ordinary Least Squares so widely used?
Ordinary Least Squares is widely used due to its simplicity, interpretability, and effectiveness, particularly in linear regression. W¹⁴hen its underlying assumptions are met, OLS produces unbiased and efficient coefficient estimates, making it a reliable tool for understanding relationships between variables and making predictions.

¹², ¹³### Can Ordinary Least Squares be used for non-linear relationships?
Ordinary Least Squares is inherently designed for linear relationships between the dependent variable and the independent variables. W¹⁰, ¹¹hile it cannot directly model intrinsically non-linear relationships, transformations of variables can sometimes be applied to make them linear in their parameters, allowing OLS to be used. F⁹or truly complex non-linear patterns, other regression techniques might be more appropriate.

⁷, ⁸### What is the error term in Ordinary Least Squares?
The error term ((\epsilon)) in an Ordinary Least Squares regression model represents the unpredictable, random variation in the dependent variable that cannot be explained by the independent variables included in the model. I⁵, ⁶t captures all other factors, unobserved or unmeasured, that influence the dependent variable, as well as random noise. O⁴LS aims for these errors to have a mean of zero, constant variance, and no correlation with other errors or independent variables.¹, ², ³